[R-sig-ME] Results from vcov() in lme4.

[Phillip Alday]

The MixedModels.jl documentation has the technical answer as to what's
going on:

> The distinction between the "fast" and "slow" algorithms in the MixedModels package (nAGQ=0 or nAGQ=1 in lme4) is whether the fixed-effects parameters, β, are optimized in PIRLS or in the nonlinear optimizer. 

My suspicion is that not doing joint optimization (i.e. estimating FE
and RE separately as in nAGQ=0) leads to the parameter space not being
explored as efficiently so you still closer to the starting values.

For GLMM this matters because of the way the fixed effects are
conditional on the RE. This was discussed a bit on the list a while
back: GLMMadaptive, if I recall correctly, can produce both conditional
and marginal (population-level) effect estimates for binomial models.
For LMM, the conditional and marginal estimates work out to be the same
thing, so you can ignore / marginalize out the fixed effects before
using the non-linear optimizer to solve the RE.

My explanation is surely infelicitous in some way, so more knowledgeable
people please correct me!

Phillip

On 12/8/19 11:40 am, Rolf Turner wrote:
> 
> I am rather puzzled by the results of applying vcov() to (binomial)
> models fitted by glmer().  The linear predictor in the models is of the
> form
> 
>      alpha_i + beta_i * x + <random effects>
> 
> where "i" corresponds to "treatment group" and x is numeric.  There are
> six treatment groups so the covariance matrix returned by vcov() is
> 12 x 12.
> 
> When I fit the model with nAGQ = 0 in the call to glmer() the resulting
> matrix is quite sparse --- there is non-zero covariance only between
> parameter estimates corresponding to the same value of "i" (i.e. to the
> same treatment group).  In other words, under the appropriate ordering
> of the estimated parameters, the covariance matrix is block diagonal,
> with six 2 x 2 blocks.
> 
> When I fit the model with nAGQ set equal to 1, the resulting covariance
> matrix is non-sparse; all entries are non-zero.  (The entries outside of
> the "block diagonal structure" are, I suppose, "relatively" small ---
> they range from -0.0145 to 0.0010 --- but are well away from being
> "approximately zero".)
> 
> I would like a better understanding of the reason for the difference.  I
> was under the impression that setting nAGQ = 0 gives a somewhat
> quick-and-dirty fit of the model; less accurate and reliable than with
> nAGQ = 1.  Why does setting nAGQ = 0 (apparently) cause the covariance
> between parameter estimates corresponding to different treatment groups
> to be *exactly* zero?
> 
> If I remember my childhood teaching correctly, in a *linear* model, such
> covariances would indeed be exactly zero, so one might expect
> "approximately" zero in the generalised linear model setting.  But why
> should setting nAGQ = 0 result in a covariance matrix which is "just
> like" one from the linear model setting?
> 
> I guess it doesn't really matter a damn, but I'd like to understand what
> is going on, at least "in rough intuitive terms".
> 
> Can anyone enlighten me?
> 
> Thanks.
> 
> cheers,
> 
> Rolf Turner
>